What Is a Circle in Coordinate Geometry?

How the graph and formula agree

A circle can also be viewed as a conic section, produced when a plane cuts a cone perpendicular to its axis. In coordinate work, however, the center-radius description is usually the most practical starting point.

This page keeps the center, radius, and equation-linked graph together so you can see how a familiar geometric figure becomes an algebraic curve without losing its meaning.

In (x - 3)² + (y + 2)² = 25, the center is (3, -2) and the radius is 5. The sign inside each bracket is opposite the coordinate of the center. Plotting that center first and drawing five units out in several directions is a reliable check on the equation.

• A circle is a conic section consisting of all points a fixed distance from one center.
• The standard center-radius equation records equal distance from the center in algebraic form.
• A circle is the special ellipse whose horizontal and vertical radii are equal.
• Changing the center shifts the graph, while changing the radius changes the size of the circle.

Where the coordinate circle helps

• Use the analytic circle when graphing center-radius equations or writing an equation from a graph.
• Use it in coordinate proofs involving fixed distance from a point.
• Use it as the conic model behind many geometry, physics, and design problems involving round paths.

Circle-equation checks

• Do not confuse the center coordinates with the radius value when reading the equation.
• Do not forget that radius is a distance and must be nonnegative.
• Do not mistake an ellipse with unequal axes for a circle just because it looks round on a stretched graph.